Subtraction of Unlike Fractions

We will learn how to solve subtraction of unlike fractions. In order to subtract unlike fractions first we convert them as like fractions.

To subtract unlike fractions, we first convert them into like fractions. In order to make a common denominator, we find LCM of all the different denominators of given fractions and then make them equivalent fractions with a common denominators.

Let us consider some of the examples of subtracting unlike fractions:

1. Subtract 1/10 from 2/5.

Solution:

2/5 - 1/10

The L.C.M. of the denominators 10 and 5 is 10.

2/5 = (2 × 2)/(5 × 2) = 4/10, (because 10 ÷ 5 = 2)

1/10 = (1 × 1)/(10 × 1) = 1/10, (because 10 ÷ 10 = 1)

Thus, 2/5 - 1/10

= 4/10 - 1/10

= (4 - 1)/10

= 3/10


2. Subtract \(\frac{3}{8}\) from \(\frac{5}{12}\).

Solution:

Let us find the LCM of denominators 8 and 12. LCM is 24.

\(\frac{3}{8}\) = \(\frac{3 × 3}{8 × 3}\) = \(\frac{9}{24}\) and

\(\frac{5}{12}\) = \(\frac{5 × 2}{12 × 2}\) = \(\frac{10}{24}\)

Now, subtract \(\frac{9}{24}\) and \(\frac{10}{24}\).

\(\frac{10}{24}\) - \(\frac{9}{24}\)                                    

= \(\frac{10 - 9}{24}\)

= \(\frac{1}{24}\)

Let us illustrate the above example pictorially as shown below.

Subtraction of Fractions

The whole strip above has 24 equal parts. The fraction \(\frac{5}{12}\) is equal to \(\frac{10}{24}\). So the shaded portion represents \(\frac{10}{24}\). We take away \(\frac{3}{8}\) or \(\frac{9}{24}\) of the above strip. The remaining part represents \(\frac{1}{24}\) of the whole strip.


3. Subtract 4/9 from 5/7.

Solution:

5/7 - 4/9

The L.C.M. of the denominators 9 and 7 is 63.

5/7 = (5 × 9)/(7 × 9) = 45/63, (because 63 ÷ 7 = 9)

4/9 = (4 × 7)/(9 × 7) = 28/63, (because 63 ÷ 9 = 7)

Thus, 5/7 - 4/9

= 45/63 - 28/63

= (45 - 28)/63

= 17/63


4. Subtract 5/8 from 1.

Solution:

1 - 5/8

= 1/1 - 5/8

The L.C.M. of the denominators 1 and 8 is 8.

1/1 = (1 × 8)/(1 × 8) = 8/8, (because 8 ÷ 1 = 8)

5/8 = (5 × 1)/(8 × 1) = 5/8, (because 8 ÷ 8 = 1)

Thus, 1/1 - 5/8

= 8/8 - 5/8

= (8 - 5)/8

= 3/8

 

5. Subtract 19/36 from 23/24.

Solution:

23/24 - 19/36

The L.C.M. of the denominators 24 and 36 is 72.

23/24 = (23 × 3)/(24 × 3) = 69/72, (because 72 ÷ 24 = 3)

19/36 = (19 × 2)/(36 × 2) = 38/72, (because 72 ÷ 36 = 2)

Thus, 23/24 - 19/36

= 69/72 - 38/72

= (69 - 38)/72

= 31/72


6. Subtract 9/35 from 3/7.

Solution:

3/7 - 9/35

The L.C.M. of the denominators 7 and 35 is 35.

3/7 = (3 × 5)/(7 × 5) = 15/35, (because 35 ÷ 7 = 5)

9/35 = (9 × 1)/(35 × 1) = 9/35, (because 35 ÷ 35 = 1)

Thus, 3/7 - 9/35

= 15/35 - 9/35

= (15 - 9)/35

= 6/35 

Subtraction of Unlike Fractions


7. Subtract \(\frac{2}{5}\) from 7.

Solution:

\(\frac{7}{1}\) - \(\frac{2}{5}\)

= \(\frac{7  × 5 - 2 × 1}{5}\) LCM of 1 and 5 is 5

= \(\frac{35 -2}{5}\)

= \(\frac{33}{5}\)

= 6\(\frac{3}{5}\)

Hence, 7 - \(\frac{2}{5}\) = 6\(\frac{3}{5}\)


Note: We write the whole number in the fraction form by keeping 1 in the denominator.


Subtraction of Fractions having the Different Denominator:

8. Subtract \(\frac{2}{3}\) - \(\frac{1}{4}\)

\(\frac{2}{3}\) = \(\frac{8}{12}\) [\(\frac{2 × 4}{3 × 4}\) = \(\frac{8}{12}\)]

\(\frac{1}{4}\) = \(\frac{3}{12}\) [\(\frac{1 × 3}{4 × 3}\) = \(\frac{3}{12}\)]

\(\frac{2}{3}\) - \(\frac{1}{4}\) = \(\frac{8}{12}\) - \(\frac{3}{12}\)

= \(\frac{8 - 3}{12}\)

= \(\frac{5}{12}\)

Method 1:

Step I: Find the L.C.M. of the denominators 3 and 4.

L.C.M. of 3 and 4 is 12

Step II: Write the equivalent fractions of \(\frac{2}{3}\) and \(\frac{1}{4}\) with denominator 12.

Step III: Subtract

Step IV: Write the difference in lowest terms.


9. Subtract \(\frac{5}{6}\) - \(\frac{1}{8}\)

\(\frac{5}{6}\) - \(\frac{1}{8}\) = \(\frac{(24 ÷ 6) × 5 – (24 ÷ 8) × 1}{24}\)

= \(\frac{(4 × 5) – (3 × 1)}{24}\)

= \(\frac{20 - 3}{24}\)

= \(\frac{17}{24}\)


Method 2:

L.C.M. of 6 and 8


Subtraction of Mixed Numbers:

Method I:

Subtract 8\(\frac{1}{2}\) - 3\(\frac{1}{4}\)

8\(\frac{1}{2}\) - 3\(\frac{1}{4}\) = (8 – 3) + [\(\frac{1}{2}\) - \(\frac{1}{4}\)]

= 5 + [\(\frac{1}{2}\) - \(\frac{1}{4}\)]

= 5 + [\(\frac{2}{4}\) - \(\frac{1}{4}\)]

= 5 + \(\frac{1}{4}\)

= 5\(\frac{1}{4}\)

Method II:

Subtract 8\(\frac{1}{2}\) - 3\(\frac{1}{4}\)

L.C.M. of 4 and 2 is 4.

8\(\frac{1}{2}\) - 3\(\frac{1}{4}\) = \(\frac{17}{2}\) - \(\frac{13}{4}\)

= \(\frac{34}{4}\) - \(\frac{13}{4}\)

= \(\frac{34 - 13}{4}\)]

= \(\frac{21}{4}\)

= 5\(\frac{1}{4}\)


2. What is 1\(\frac{4}{5}\) less than 4\(\frac{1}{2}\)?

Find 4\(\frac{1}{2}\) - 1\(\frac{4}{5}\)

4\(\frac{1}{2}\) - 1\(\frac{4}{5}\) = \(\frac{9}{2}\) - \(\frac{9}{5}\)            L.C.M. of 2 and 5 is 10.

             = \(\frac{45}{10}\) - \(\frac{18}{10}\)

             = \(\frac{45 - 18}{10}\)

             = \(\frac{27}{10}\)

            = 2\(\frac{7}{10}\)



Questions and Answers on Subtraction of Unlike Fractions:

1. Find the difference:

(i) \(\frac{3}{8}\) - \(\frac{1}{8}\)

(ii) \(\frac{17}{23}\) - \(\frac{6}{23}\)

(iii) \(\frac{1}{2}\) - \(\frac{3}{16}\)

(iv) \(\frac{5}{14}\) - \(\frac{2}{7}\)

(v) \(\frac{5}{6}\) - \(\frac{3}{4}\)

(vi) \(\frac{2}{3}\) - \(\frac{1}{5}\)

(vii) 5 - \(\frac{3}{4}\)

(viii) 2 - \(\frac{15}{21}\)

(ix) 4\(\frac{2}{3}\) - 2



Answers:

1. (i) \(\frac{1}{4}\)

(ii) \(\frac{11}{23}\)

(iii) \(\frac{5}{16}\)

(iv) \(\frac{1}{14}\)

(v) \(\frac{1}{12}\)

(vi) \(\frac{7}{15}\)

(vii) \(\frac{17}{4}\)

(viii) \(\frac{27}{21}\)

(ix) 2\(\frac{2}{3}\)


2. Subtract the following Unlike Fractions:

(i) \(\frac{4}{7}\) - \(\frac{1}{3}\)

(ii) \(\frac{3}{4}\) - \(\frac{1}{2}\)

(iii) 8 - \(\frac{2}{3}\)

(iv) 1\(\frac{5}{6}\) - 1\(\frac{1}{2}\)

(v) 4\(\frac{3}{4}\) - \(\frac{1}{2}\)

(vi) 2\(\frac{1}{3}\) - 1\(\frac{1}{2}\)

(vii) 13\(\frac{4}{7}\) - 6

(viii) 7\(\frac{2}{5}\) - 3\(\frac{1}{2}\)

(ix) \(\frac{9}{2}\) - 4

(x) \(\frac{2}{5}\) - \(\frac{3}{10}\)


Answer: 

2. (i) \(\frac{5}{21}\)

(ii) \(\frac{1}{4}\) 

(iii) 7\(\frac{1}{3}\)

(iv) \(\frac{1}{3}\)

(v) 4\(\frac{1}{4}\)

(vi) \(\frac{5}{6}\)

(vii) 7\(\frac{4}{7}\)

(viii) 3\(\frac{9}{10}\)

(ix) \(\frac{1}{2}\)

(x) \(\frac{1}{10}\)

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